continuous functions on the extended real numbers

Theorem 1.

is continuous if and only if f is continuous such that limx→∞⁡f⁢(x)=A and limx→-∞⁡f⁢(x)=B for some A,B∈R¯.

Proof.

Note that f¯ is continuous if and only if limx→c⁡f¯⁢(x)=f¯⁢(c) for all c∈ℝ¯. By defintion of f¯ and the topology of ℝ¯, limx→c⁡f¯⁢(x)=limx→c⁡f⁢(x) for all c∈ℝ¯. Thus, f¯ is continuous if and only if limx→c⁡f⁢(x)=f¯⁢(c) for all c∈ℝ¯. The latter condition is equivalent (http://planetmath.org/Equivalent3) to the hypotheses that f is continuous on ℝ, limx→∞⁡f⁢(x)=A, and limx→-∞⁡f⁢(x)=B.
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